Newton Method for Multiobjective Optimization Problems of Interval-Valued Maps

In this article, we propose a Newton-based method for solving multiobjective interval optimization problems (MIOPs). We first provide a connection between weakly Pareto optimal points and Pareto critical points in the context of MIOPs. Introducing this relationship, we develop an algorithm aimed at computing a Pareto critical point. The algorithm incorporates the computation of a descent direction at a non-Pareto critical point and employs an Armijo-like line search strategy to ensure sufficient decrease. Under suitable assumptions, we prove that the sequence generated by our proposed algorithm converges to a Pareto critical point. The effectiveness and performance of the proposed method are demonstrated through a series of numerical experiments on some test problems. Finally, we apply our proposed algorithm in a portfolio optimization problem with interval uncertainty.

💡 Research Summary

The paper addresses the challenging class of multi‑objective optimization problems whose objective functions are interval‑valued (MIOPs). Recognizing that real‑world decision making often involves both multiple conflicting criteria and uncertain data, the authors develop a Newton‑type algorithm that works directly on interval‑valued maps (IVMs) without converting them into separate lower‑ and upper‑bound scalar problems.

The first theoretical contribution is a rigorous connection between weak Pareto optimality and Pareto criticality in the interval setting. By defining a suitable dominance relation for intervals and employing the generalized Hukuhara (gH) difference, the authors show that every weakly Pareto optimal point is a Pareto critical point under their definitions, establishing a solid foundation for algorithmic development.

The algorithm proceeds as follows. At a current iterate (x^k) that is not Pareto critical, the gH‑gradient (\nabla_{gH}F(x^k)) and the gH‑Hessian (\nabla^2_{gH}F(x^k)) of the vector‑valued interval map (F) are computed. Using these, a multi‑objective Newton system is formed: a strongly convex quadratic subproblem whose solution yields a descent direction (d^k) that simultaneously reduces all interval objectives according to the interval dominance order. The direction is obtained by solving a linear system involving the gH‑Hessian, which is shown to be positive definite under the standing assumptions.

A line‑search based on an Armijo‑type condition is then applied. The authors prove the existence of a step length (\alpha_k>0) that satisfies the sufficient decrease condition for the interval‑valued merit function, ensuring that the new iterate (x^{k+1}=x^k+\alpha_k d^k) yields a strict improvement in the sense of interval dominance.

Convergence analysis assumes that each component’s lower and upper bound functions are twice continuously differentiable, that the gH‑gradient is Lipschitz continuous, and that the gH‑Hessian is uniformly positive definite in a neighbourhood of the solution set. Under these conditions, the generated sequence ({x^k}) remains bounded, every accumulation point is a Pareto critical point, and the algorithm is shown to be scaling‑independent (i.e., invariant under affine transformations of the decision variables).

Numerical experiments are conducted on a suite of benchmark MIOPs (ranging from two to four objectives) and on a realistic portfolio optimization problem where asset returns and risks are modeled as intervals. The proposed Newton method is compared against a previously published steepest‑descent algorithm for MIOPs and against a transformation‑based Newton/quasi‑Newton approach that treats lower and upper bounds separately. Results demonstrate that the new method converges in fewer iterations, attains a more extensive approximation of the Pareto front, and captures Pareto points that are missed by the transformation approach because it exploits the full interval information rather than a scalarized surrogate. In the portfolio case, the algorithm efficiently identifies efficient frontiers that balance expected return and risk under interval uncertainty, offering more robust investment strategies.

The paper concludes by highlighting its contributions: a novel theoretical framework linking weak Pareto optimality to Pareto criticality for interval‑valued maps, a Newton algorithm that directly utilizes gH‑gradient and gH‑Hessian information, and a convergence proof under realistic smoothness assumptions. Future research directions suggested include extending the method to constrained MIOPs, handling non‑convex interval objectives, developing global convergence guarantees, and scaling the algorithm to high‑dimensional problems through approximated Hessians or parallel implementations. Overall, the work significantly advances the state‑of‑the‑art in multi‑objective interval optimization by providing both rigorous analysis and practical computational tools.